1. Solving Systems of Equations by Fen Xu and Timothy Lou Ly f(x)= 2x 2 - 2f(x)= -1/2x 2 + 18

2. The Concept Graphing a Linear Equation y = 1x X-ValuesY-Values -24 -20 -16 -12 -8 y = 2x - 4 X-ValuesY-Values 812 914 1016 1118 1220 (-20, -20) (-16, -16) (-8, -8) (8, 12) (10, 16) (12, 20)

3. The Concept Dependent SystemInconsistent System Two overlapping lines with the same slope and points Two lines with the same slopes that never intersect or share points.

4. The Concept Three Ways to Solve - Graphing - Addition - Substitution

5. Addition = 3x - 3 + 2y + 4 = 24- 3+ 4 Step 1: Make equations into simplest form of Ax + By = C. = 3x + 2y = 23 Step 2: Choose to solve for “x” or “y.” For the example, we’ll solve for “x.” Multiply by LCD to get rid of fractions Combine like terms by adding -3 and +4 together 6 ( ) _ 2 3 x - 1 y + 2 _ += 4 x - 2y = 5 simplest form Ax + By = C solve for “x” or “y.” = 3x + 2y - 1 = 24 24 Add +1 to both sides to cancel -1 and isolate variables

6. Addition Step 3: Add the two equations so that the y-values cancel. 3x + 2y = 23 + ( x - 2y = 5) _ Because 2y is being subtracted from 2y, they cancel 4x = 28 Step 4: Continue to solve for “x.” Divide both sides by 4 x = 7 Addequations soy-values solve for “x.” / / 4 If you wanted to solve for “y” and cancel “x,” you would need to multiply the 2nd equation by -3 cancel.

7. Addition Step 5: We can now put 7 in for “x” in any equation and find the value of “y.” put 7 in for “x” find“y.” x - 2y = 5 = 7 - 2y = 5 = -2y = -2 = y = 1 CHECK: (7) - 2(1) = 5 (7) - 2 = 5 5 = 5 √ √ That’s it! (7, 1) is your solution/intersection! -2 3x + 2y = 23x- 2y = 5 3(7) + 2(1) = 23 21 + 2 = 23 23 = 23

8. Substitution Step 1: Solve for one of the variables from one equation. Solve for onevariablesone equation = 3x + 2y = 233x = 2y = -3x + 232 Subtract 3x from both sides Divide the equation by 2 _ -3 2 = y = x 11.5 + Multiply by LCD to get rid of fractions 6 ( ) x - 2y = 5 _ 2 3 x - 1 y + 2 _ += 4 = 3x + 2y - 1 = 24 24 Add +1 to both sides to cancel -1 and isolate variables = 3x - 3 + 2y + 4 = 24- 3+ 4 Combine like terms by adding -3 and +4 together

9. _ -3 2 x 11.5 + Substitution Step 2: Substitute “ ” for “y” in the other equation: x - 2y = 5 and solve. x - 2( ) = 5- 2() = x + 3x - 23 = 5 - 235 = 4x = 28 4 = x = 7 Multiply out -2 Add 23 to both sides Divide the equation by 4 Step 3: Putting in 7 for “x,” we know that y = 1. x - 2y = 5= 7 - 2y = 5= -2y = -2 = y = 1-2 7 for “x,”y = 1 _ -3 2 y = x 11.5 +

10. X-ValuesY-Values 011.5 28.5 45.5 62.5 71 X-ValuesY-Values 0-2.5 2-1.5 4-0.5 60.5 71 _ 2 -3 y = x 11.5+ _ 1 2 y = x2.5- (0, 11.5) (4, 5.5) (4, -0.5) (0, -2.5)

11. Review 7x-6y=-6 -7x+6y=-4 Try to solve this with the method of your choice: We’ll check, and if you get it right, you get some candy! (You’ll all get some anyways)